Two functions are homotopic, if one of them can by continuously deformed to another. This gives rise to an equivalence relation. A group called homotopy group can be obtained from the equivalence classes. The simplest homotopy group is fundamental group. Homotopy groups are important invariants in ...

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4
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0answers
31 views

bijection between the free homotopy classes $[S^{n},X]$ and the orbit space

I would like to prove that there is a bijection between the free homotopy classes $[S^{n},X]$ and the orbit space $\pi_{n}(X,x_{0}) / \pi_{1}(X,x_{0})$ where the action of $\pi_{1}(X,x_{0})$ over ...
0
votes
1answer
40 views

relative homotopy groups

I study relative homotopy groups and I have a question: Let $A\subseteq X$ (not necessarily CW complex) and $\pi_{n}(X,A)$. Is it always possible to find a pointed space Y for which ...
4
votes
0answers
27 views

Transversality and homotopic maps

I'm trying to solve some problems in differential topology, and I came across the following: suppose $f:M\times [0,1]\rightarrow N$ is a homotopy, where $M$ is a compact manifold, such that $f_0$ and ...
4
votes
1answer
71 views

If $f:S^1\to S^1$ doesn't have any fixed point then it is homotopic to the identity

How to show that every continuous function $f:S^1\to S^1$ without fixed points is homotopic to the identity? (without using homology nor the concept of degree).
6
votes
1answer
90 views

Homotopic equivalence for $S^5$ without three $S^1$

Consider a standard embedding of $S^5$ in $\mathbb R^6$: $S^5: \; x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 = 1.$ And consider three circles, which are sections of $S^5$ by $x_1 x_2$, $x_3 x_4$, ...
9
votes
3answers
971 views

An intuitive idea about fundamental group of $\mathbb{RP}^2$

Someone can explain me with an example, what is the meaning why $\pi(\mathbb{RP}^2,x_0)$ is $\mathbb{Z}_2$? We consider the real projective plane as a quotient of the disk. I don't receive and ...
3
votes
1answer
49 views

Does the functor $\mathbf{cosk_n}:sSet\to sSet$ preserve Kan complexes?

On this nlab page a functor $\mathbf{cosk_n}:sSet\to sSet$ is constructed. Is $\mathbf{cosk_n}(K)$ of a Kan complex $K$ again a Kan complex?
0
votes
0answers
30 views

Question on cofiber sequence map in equivariant homotopy theory

Let $G=C_2$ denote the cyclic group with two elements. Up to isomorphism there are only two irreducible $C_2$-representations, the identity representation, $\mathbb{R}^{1,0}$, and the sign ...
1
vote
1answer
25 views

Let $X\subset\mathbb{R}^3$ be the union of the coordinate axies, I want to show that $\mathbb{R}^3-X$ is homotopy equivalent to a graph

Let $X\subset\mathbb{R}^3$ be the union of the coordinate axies, I want to show that $\mathbb{R}^3-X$ is homotopy equivalent to a graph, and the question asks further "which graph" Let ...
0
votes
1answer
96 views

How to show $S^n$ is not contractible without using Homology..

I know the prove $S^n$ is not contractible using homology.But I don't know how to prove it from definition of contractibility.Can anyone help me in this direction? Thanks.
1
vote
2answers
55 views

Show that the Möbius band has its central circle $C$ as a deformation retract

I have started this problem by using the planar representation of the Möbius band and noted that a line down the middle is probably what is meant by the central circle, since travelling from top to ...
0
votes
1answer
29 views

Simple homotopy construction

I'm sure this isn't too difficult but i can't seem to do it if you have two loops $p_0 = e*g $ and $p_1 = g*e$ where $e$ is the trivial loop How would i construct an explicit homotopy between the ...
4
votes
1answer
54 views

Relationship between cohomology and higher-homotopy

Let $M$ be a connected, compact, and orientable 3-manifold ($H^3(M)\cong\mathbb{Z}$), and let $G$ be a simple Lie group satisfying $\pi_1(G)=\pi_2(G)=0$. Let $\pi_M(G)$ denote the set of homotopy ...
1
vote
0answers
17 views

Tools for proving map is homotopy equivalence

General situation I'm preparing a geometry exam, and a lot of exercises from past years' exams are of the form «given $f$ the map so-and-so, prove (or determine whether) it is a homotopy ...
4
votes
3answers
70 views

The significance of filtered colimits in homotopy theory

I have been allowed to attend some preparatory lectures for a seminar on the Goodwillie Calculus of Functors. I found in my notes from one of the lectures two statements which I would like to ask ...
-1
votes
0answers
10 views

definition of a fiberwise homotopy

I google it but I did not succeed to find a precise definition of a fiberwise homotopy. Can someone give me the definition of a fiberwise homotopy ? Thanks
5
votes
1answer
95 views

What does it mean for a category to be “tensored over” another category?

What does it mean for a category to be "tensored over" another category? I was reading "Stable model categories are categories of modules" by Schwede and Shipley ...
4
votes
1answer
60 views

definitions of various spectra: $E^X$ and $E \wedge \Sigma^\infty X$

Let $E=\{E_n\}$ be a spectrum given by a sequence of pointed CW complexes $E_n$ and inclusions $\Sigma E_n \to E_{n+1}$. Let $X$ a pointed CW complex. I had a few very naive questions I had while ...
5
votes
3answers
331 views

mapping homotopic to the identity map

Please give me a hand with this problem, It was on my exam, and I just couldn't solve it. Suppose $\phi:\mathbb{S}^2\to\mathbb{S}^2$ is a mapping, homotopic to the identity map. Show that there is ...
5
votes
1answer
91 views

Finding a good cover such that its lifting is still a good cover

Let $Y$ be a compact manifold, $X$ a topological space and $f: X \to Y$ a surjective map. Suppose further that every point in $Y$ has arbitrarily small open neighbourhoods such that their preimages in ...
1
vote
1answer
45 views

Question about homotopic functions and homotopy classes

If $X = [0,1] \times [0,1]$ and $Y = [0,1] \cup [2,3]$. I have to give an example of two continuous functions $f$ and $g$ that are not homotopic. I was thinking of $f(x,y) = x$ and $g(x,y) = y +2$ ...
1
vote
0answers
26 views

functors with a morphism lifting property

By analogy to the familiar situation in homotopy theory (i.e., (unique) path lifting in covering spaces), it is natural to consider the following. Let $P:C\to D$ be a functor. Say that $P$ has ...
3
votes
1answer
360 views

local homology group

The question is: $X\in \mathbb{R}^{n}$ is the subspace ${(x_{1},...,x_{n}\mid x_{n}\geq 0)}$, and let $Y$ is the subspace with $x_{n}=0$. let $x\in Y$, calculate the local homology of $X$ at $x$. ...
1
vote
1answer
16 views

Relative homotopy and composition of maps

I am trying to prove something and am stuck on the following issue : Suppose $\Psi, \Phi : I^n \to Y$ are two maps and $q:Y \to Z$ is a homotopy equivalence such that $q \Phi \cong q \Psi $rel ...
3
votes
0answers
136 views

A construction with homotopy colimits and homotopy pullbacks for descent.

I have some troubles in trying to give a meaningful interpretation to the following property which is stated in this preprint by professor Rezk (see Definition 6.5) as part of the requirement for a ...
2
votes
1answer
55 views

Find $f$ and $g$ homotopic s.t. induce different homomorphisms

Let $X$ and $Y$ be two topological spaces. Are there continuous functions $f,g:X\to Y$ satisfying the following conditions? $f(a)=g(a)=b$ for some $a\in X$, $f$ and $g$ are homotopic, and the ...
4
votes
0answers
60 views

What is this relation on the set of paths called in graph theory?

Suppose I have a directed simple graph $\Gamma$ (no edge loops or multi-edges) and a directed path $v_0v_1\cdots v_k$ joining vertex $s=v_0$ to vertex $t=v_k$. By directed path I mean that each pair ...
3
votes
2answers
92 views

How to construct $K(\mathbb{Z}/5\mathbb{Z},1)$?

From the Wikipedia article on Eilenberg-MacLane spaces: A $K(G, n)$ can be constructed stage-by-stage, as a CW complex, starting with a wedge of n-spheres, one for each generator of the group ...
1
vote
0answers
17 views

Calderón–Zygmund lemma and Lebesgue measure

Can someone please explain the relationship between the Calderón–Zygmund lemma and Lebesgue measure, thanks.
6
votes
1answer
1k views

Using Homotopy to solve system of nonlinear equations

So far I have been using Newton-Raphson (N-R) to solve nonlinear systems. However N-R might run into the problem of singularity depending on the initial guess. I found an alternate approach which is ...
0
votes
1answer
11 views

proof of the decomposition of a fibration by a trivial fibration followed by a minimal fibration

In Hovey's book theorem 3.5.9 asserts that any fibration $p:X\rightarrow Y$ can be factored as a trivial fibration followed by a minimal fibration. The proofs begins by defining a set T of simplices ...
3
votes
3answers
66 views

Reference: In every free homotopy class is a unique minimizing closed geodesic

Does anyone know a reference for the following result: Let $M$ be a compact hyperbolic manifold/manifold with strict negative curvature . Then in every non-trivial free homotopy class of $M$ there ...
1
vote
1answer
36 views

A quotient map $X\to X/A$ that is not a Serre fibration

What is an example of a CW-pair $(X,A)$ such that the quotient map $X\to X/A$, i.e. the map obtained from the pushout \begin{eqnarray} A &\to& X\\ \downarrow &&\downarrow\\ * ...
0
votes
1answer
29 views

Based on the Andreotti-Frankel theorem, what is the CW complex homotopy equivalent to $x^2 + y^2 - 1$?

I am referring to this theorem: http://en.wikipedia.org/wiki/Andreotti%E2%80%93Frankel_theorem I have no idea how to begin thinking about this.
0
votes
1answer
36 views

Homotopy proof of Cauchy's Theorem

The proof of Cauchy's theorem in these notes http://people.math.gatech.edu/~cain/winter99/ch5.pdf rely on the concept of homotopy. But it seems to me that the proof did not use any property special to ...
2
votes
1answer
36 views

Showing that identity and g are not homotopic (without Homology)

Question: Are the identity mapping on $S^1$ and the reflection about the $x$-axe homotopic? This is a question which I already know the answer. The objective is to find better answers and suggestions ...
8
votes
1answer
166 views

Difference between homotopy equivalence and homeomorphism - dimensionality

(The most voted answer to) This question shows spaces of the same dimension can be homotopy equivalent but no homeomorphic. On the other hand "difference in dimension" is still a nice way to tell ...
5
votes
3answers
556 views

Eckmann-Hilton and higher homotopy groups

How does the Eckmann-Hilton argument show that higher homotopy groups are commutative? I can easily follow the proof on Wikipedia, but I have no good mental picture of the higher homotopy ...
1
vote
1answer
43 views

Homotopy and Semidirect Product

I know there is a relation in homotopy theory which is $\pi(G\times H) = \pi(G)\times\pi(H)$. However, does this relation still hold for $\pi_0$ which may not be a group? Moreover, is there such a ...
2
votes
1answer
69 views

When a homotopy equivalence of the closed unit ball in the Euclidean n- dimensional space is injective?

I have a compact connected metric space $X$ of dimension $n$ which is homotopically equivalent to the closed unit ball $D^n$ in the n-dimensional Euclidean space. I am wondering if there is an ...
-1
votes
1answer
33 views

Homotopy split monomorphisms [closed]

Let $C$ be a model category. Recall that a morphisme $f : X \to Y$ is called a homotopy monomorphism if the diagonal $X \to X \times^h_Y X$ induces an isomorphism in the homotopy category. Suppose ...
4
votes
1answer
100 views

Utility of the 2-Categorical Structure of $\mathsf{Top}$?

It's well known that $\mathsf{Top}$ is a 2-category with homotopy classes of homotopies as 2-arrows. I'm a bit afraid to ask this question, but what is the utility of this 2-categorical structure? ...
3
votes
1answer
48 views

Weak equivalence testable on invariant open covers?

Let $f\colon X\rightarrow X'$ be a continuous map between two spaces $X,X'$, which might be arbitrary wild, especially I don't want to work in any convenient category of topological spaces. Let ...
1
vote
1answer
37 views

Question on showing a bijection between $\pi_1(X,x_0)$ and $[S^1, X]$ when X is path connected.

I am trying to do this question taken from Hatchers algebraic topology and I am struggling to understand the notation and the concepts. As far as I know $\pi_1(X,x_0)$ is the set of end point ...
4
votes
0answers
88 views

Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general situation. Especially I do not work in any "convenient" category of spaces. Wikipedia ...
2
votes
1answer
709 views

cauchy theorem over cycles homologous to zero

Definitions: $i)$ A cycle $\gamma$ is a finite sequence of continuous oriented closed paths in the complex plane. We denote $\gamma = (\gamma_1,...\gamma_n)$ where $\gamma_k$ are the closed paths of ...
0
votes
1answer
34 views

Simplicial approximation

One of the definition of simplicial approximation says that: a simplicial map $h:|K|\rightarrow|L|$ is a simplicial approximation of a continuous map $f:|K|\rightarrow|L|$ if and only if $$\forall ...
3
votes
1answer
52 views

Aspherical but not contractible

Let $X$ be the topologist's sine curve (i.e. $\left\lbrace (x,y): y=\sin\left(\frac{1}{x}\right),x\in ]0,1]\right\rbrace\cup \lbrace (0,y): y\in [-1,1]\rbrace$) with an arc joining $(0,0)$ and ...
1
vote
0answers
24 views

Criterion for projectively cofibrant diagrams?

Let $\mathbf{D}$ be a small category, $\mathbf{Set}$ the category of sets and $\mathbf{sSet}$ the category of simplicial sets with its usual model structure. The category of functors $\mathbf{D}\to ...
2
votes
2answers
56 views

Residue Theorem and Homologous to zero

This is a very basic question and I couldn't find it posted yet but here it goes; The Residue Theorem states that if $f:G\to \mathbb{C}$ is analytic on $G$- a region and $f$ has isolated singularities ...